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37 Proof.By Lemma 2.1 of Ročková (2023), For any priorπ(θ), ρ(θ,ˆp) =θ2 2r−EθlogNθ,v(Z) +E θlogNθ,1(Z),(A.6) where Nθ,v(Z) = ∫ R exp { µZ√v + µθ v −µ2 2v } π(µ) dµ, andv−1= 1 +r−1

Specifically, ifθfollows Horseshoe prior with a fixedτ >0, then ρ(θ,ˆp) =θ2 2r−EθlogN HS θ,v(Z) +E θlogN HS θ,1(Z),(A · 2014

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Horseshoe Predictive Inference

math.ST · 2026-04-17 · unverdicted · novelty 7.0

The Horseshoe prior yields asymptotically minimax optimal predictive Bayes estimators in sparse Gaussian models, with hierarchical extensions enabling adaptive switching and sharper risk bounds under a theta-min condition.

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  • Horseshoe Predictive Inference math.ST · 2026-04-17 · unverdicted · none · ref 2

    The Horseshoe prior yields asymptotically minimax optimal predictive Bayes estimators in sparse Gaussian models, with hierarchical extensions enabling adaptive switching and sharper risk bounds under a theta-min condition.